Hwang Ui-jo gave Bordeaux a surprise lead before Cavani equalised for PSG with his landmark strike at the Parc des Princes.
The hosts went ahead when Marquinhos chested the ball home but Bordeaux levelled through Pablo following a mistake by PSG goalkeeper Sergio Rico.
Marquinhos and Kylian Mbappe both finished clinically to put Thomas Tuchel's side in command, although Ruben Pardo reduced the deficit to set up a tense finish and Neymar was dismissed for a second booking late on.